Conformal Mapping: region exterior two to circles with common tangency at a point.

Question

I want to map the region outside two circles: $|z+2|=2$ and $|z-1|=1$ onto something easier to work; Perhaps a strip. Obviously, the two circles have a common tangent at $(0,0)$.

I will be totally honest here, I have no Idea how to proceed or else I would have shown what I have so far. Regards.

Answer 1

The region you describe is not simply connected, so there cannot exist a 1-1 holomorphic map of this region onto a strip. (The map discussed in the comments, namely $1/z,$ maps your region 1-1 onto $\{-1/4<\text {Re }z< 1/2\}\setminus \{0\}.$)